Problem Analysis:

We are tasked with determining $n(Y)$ given the following details:

1. The universal set $U$ consists of natural numbers from 1 to 3500. Thus, $n(U) = 3500$.
2. $X$ and $Y$ are subsets of $U$, and $n(X) = 975$.
3. We need to calculate $n(Y)$ or explain why it's not possible.

Solution:

The problem does not give any relationship between sets $X$ and $Y$. Specifically, it does not state if $X \cap Y$ (the intersection of $X$ and $Y$) is non-empty, disjoint, or any other relevant detail.

To calculate $n(Y)$, we would need more information, such as:

• Whether $Y$ is disjoint from $X$,
• Whether $Y$ is a subset of $X$,
• Or if $n(X \cup Y)$ (the union of $X$ and $Y$) is given.

Conclusion:

Without any additional information about $Y$ or its relationship to $X$, it is not possible to determine $n(Y)$. The given data is insufficient.

Would you like me to provide further details or explain this in a different way?

Here are 5 related questions for better understanding:

1. What is the relationship between subsets $X$ and $Y$?
2. How do we calculate the cardinality of a union or intersection of two subsets?
3. If $X$ and $Y$ were disjoint, how would this affect $n(Y)$?
4. What is the role of the universal set $U$ in problems involving subsets?
5. How does $n(U)$ relate to $n(X)$ and $n(Y)$ in general?

Tip: When solving set problems, always look for relationships between subsets like union, intersection, or complement for more insight!